As is known, short-duration impact processes like golf-club head/golf ball collisions are characterized by coefficients of restitution and friction coefficients which are computed from changes in head and ball linear and angular momenta. While changes in linear momentum are computed from the ball and head mass, changes in angular momentum require accurate knowledge of the club-head inertia matrix, alternatively referred to as the three dimensional inertia tensor. Golf-club heads, like many other irregular rigid bodies with unknown internal mass distributions, are difficult to characterize accurately by calculated inertia parameters.
Generally, little is known is about the significance to the game of golf of one shape versus another. However, there is an expectation that it is the inertia that opposes the torque resulting from an off-center hit. Thus, knowledge of the inertia of a golf-club head should aid in the design of golf-club head for a particular golf-club size.
In the past, various types of techniques have been employed for determining the swing inertia of a golf-club, for example, as described in U.S. Pat. No. 4,603,577, as well as to determine the moment of inertia of a golf-club such as described in U.S. Pat. 4,212,193. However, club head/ball motion before and after collision is generally a three-dimensional motion. As such, the angular momentum cannot be computed from a single moment of inertia as would be the case for two dimensional motion. In the case of bodies with planes of symmetry, it may be possible to compute an entire inertia matrix from three or fewer principal moments of inertia. However, in the absence of symmetry, as in the case of golf-club heads, the three moments of inertia and three products of inertia must be computed or measured. Measurement of moments of inertia about three mutually perpendicular axes can be achieved by replication of the single-axis prior art. There is no comparable way of measuring products of inertia.
If one knows the moments of inertia about each axis of an orthogonal set of axes for multiple sets of orthogonal axes, it can be possible to compute the products of inertia with respect to one of those sets of orthogonal axes. Once the moments and products of inertia are known for any one set of orthogonal axes, they can be computed for any other set of axes. The moments of inertia together with the products of inertia constitute the inertia matrix.
Accordingly, it is an object of the invention to obtain direct measurements of moments of inertia and the indirect determinations of products of inertia so as to permit a complete characterization of the motion of an irregularly-shaped rigid body.
It is another object of the invention to permit the measurement of inertial properties of rigid bodies in cases where inhomogeneity, irregularity or complexity makes computing such properties impractical.
It is another object of the invention to be able to compare the inertial properties of similar objects or the establishment of functional similarity among objects of differing appearances.
It is another object of the invention to be able to determine the inertia matrix of a rigid body in a relatively simple manner.
It is another object of the invention to be able to determine the inertia matrix of a golf-club head.
It is another object of the invention to provide a relatively simple basis for designing golf-club heads.
It is another object of the invention to obtain a graphical representation of an inertia matrix for a golf-club head against which golf-club heads can be matched.
It is another object of the invention to provide an apparatus and procedure to permit the experimental determination of a location of a center of mass and the moments and products of inertia of an irregularly shaped solid body.
Briefly, the invention provides an apparatus and method for determining the inertia matrix of a rigid body. In this respect, the apparatus and method permit an experimental determination of the location of the center of mass and the moments and products of inertia of an irregularly shaped solid body, such as a rigid golf-club head.
The apparatus is comprised of a pair of spaced apart supports, a means removably secured to the beam to receive a known weight thereon and a first adapter for mounting a club head on the balance beam for selectively balancing the club head about a Z-axis extending along a center line of the hosel of the club head relative to the known weight with a face of the club head disposed horizontally in order to determine an x-coordinate of the center of mass of the club head and with the face of the club head disposed vertically to determine a y-coordinate of the center of mass. In addition, the apparatus employs a second adapter for mounting the club head on the beam with the Z-axis perpendicular to the beam for balancing the club head about an X-axis parallel to the face of the club head in order to determine a z-coordinate of the center of mass.
The apparatus also includes a third adapter for mounting the club head on the beam in the absence of the known weight with the Z-axis extending vertically and perpendicularly to the beam in order to obtain moments of inertia I.sub.XX, I.sub.YY about the respective x and y axes of the club head.
The apparatus also includes a fourth adapter for mounting the club head on the beam with the Z-axis extending orthogonally at an angle of 45.degree. relative to the beam to obtain additional moments of inertia I.sub.x''x'', I.sub.z''z''.
The method or procedure of the invention permits measurement of the moments of inertia with respect to a base axis system and with respect to three additional axis systems. Like many bodies, golf-club heads have physical features that suggest a base axis system, e.g., the hosel face and centerline. Then the products of inertia in the base axis system can be computed, thereby completing the inertia matrix with respect to an axis system with an origin at the mass center and axes parallel to those of the base axis system. As part of the procedure, it is necessary to locate the center of mass with respect to the base axis system. The same device is adapted to this purpose.